INNER FUNCTIONS, BLOCH SPACES AND SYMMETRIC MEASURES

INNER FUNCTIONS, BLOCH SPACES AND SYMMETRIC MEASURES

A. B. ALEKSANDROV, J. M. ANDERSON and A. NICOLAU [Received 7 November 1997ÐRevised 21 July 1998]

1. Introduction

Let H¹ denote the algebra of bounded analytic functions in the unit disc D of the complex plane C. The well-known Schwarz±Pick theorem asserts that if f 2 H¹with

k f k₁ˆ supfj f …z†j: z 2 Dg < 1 then f decreases hyperbolic distances; that is,

f …z† ÿ f …a†

1 ÿ f …a† f …z†

  <

z ÿ a 1 ÿ az  

 for all z; a 2 D, or, in®nitesimally,

…1 ÿ jzj²†j f⁰…z†j < 1 ÿ j f …z†j² for z 2 D:

A function I 2 H¹ is called inner if it has radial limits of modulus 1 at almost every point of the unit circle T. If E Ì T then jEj denotes its normalized Lebesgue measure. We introduce several measures on T, but the expression

`almost every' always refers to Lebesgue measure. We assume a knowledge of inner functions, such as is to be found in [9]. In particular, we may write I as I ˆ BS where

B…z† ˆY¹

n ˆ 1

zn

jz_nj

znÿ z 1 ÿ z_nz

 

is the Blaschke product associated with the zero set fzng of I, and S ˆ S‰mŠ…z† ˆ exp ÿ

Z

T

y ‡ z y ÿ zdm…y†

 

is the singular inner factor associated with the positive singular measure m.

The ®rst result of this paper is the construction of an inner function I which, in some sense, decreases hyperbolic distances as much as desired as jzj ! 1.

Theorem 1. Let f: …0; 1Š ! …0; 1† be a continuous function with

t ! 0limf…t† ˆ 0:

The ®rst author is partially supported by the grants INTAS 93-1376 and RFFI 96-01-00693. The ®rst and second authors gratefully acknowledge the kind hospitality of the CRM at the Universitat AutoÁnoma de Barcelona during the preparation of this work. The third author is partially supported by the DGICYT grant PB95-0956 and CIRIT grant 1998SRG00052.

1991 Mathematics Subject Classi®cation: primary 30D50; secondary 30D45, 26A30, 47B38.

Proc. London Math. Soc. (3) 79 (1999) 318±352.

Then there exists an inner function I such that

j z j ! 1lim^ÿ

…1 ÿ jzj²†jI⁰…z†j f…1 ÿ jI…z†j²† ˆ 0:

We apply this theorem to prove some results on composition operators, Zygmund functions and the existence of certain singular measures.

Recall that a function f , analytic in D, is called a Bloch function if the quantity k f kBˆ supf…1 ÿ jzj²†j f⁰…z†j: z 2 Dg

is ®nite. The Banach space of all such functions is the Bloch space, denoted by B with j f …0†j ‡ k f k_B as norm. The little Bloch space B₀ is the subspace of B consisting of those f 2 B for which

j z j ! 1lim^ÿ…1 ÿ jzj²†j f⁰…z†j ˆ 0:

The Zygmund class L^ˆ L^…T† is the space of continuous functions F on T for which

supfjF…e^{i …v ‡ h†}† ‡ F…e^{i …v ÿ h†}† ÿ 2F…e^{i v}†j: e^{i v}2 Tg < K jhj

for some constant K. When the quantity above is o…jhj† as h ! 0 we say that F is in the small Zygmund class l^…T†. Roughly speaking, Zygmund functions are the primitives of functions in the Bloch space, namely an analytic function f is in B if and only if

F…z† ˆ Z _z

0 f …t† dt

belongs to L^…T† for jzj ˆ 1. Analogous relations hold between B₀ and l^ (see [18] for details).

Some consequences of Theorem 1 are as follows. Given a positive continuous function w: ‰0; 1† ! …0; ‡1† with

t ! 1lim^ÿw…t† ˆ ‡1;

let H…w† denote the Banach space of functions f , analytic in D such that k f kwˆ supfj f …z†jw…jzj†^ÿ1: z 2 Dg < 1:

Corollary 1. Let w be as above and « > 0 be given. Then there exists a non-constant inner function I such that the composition operator C…I †, de®ned as

C…I †… f † ˆ f ± I

maps H…w† into B₀. Moreover C…I † is compact with kC…I †k < «.

The argument leading from Theorem 1 to this corollary is very ¯exible and may be applied to obtain other results of a similar type. One such result is the following.

Corollary 2. Given any sequence f f_ng of analytic functions in D, there exists an inner function I such that f_n± I 2 B₀ for n ˆ 1; 2; 3; . . . :

Another application of Theorem 1 is as follows.

Corollary 3. Let I be a non-constant inner function satisfying

j z j ! 1lim^ÿ

…1 ÿ jzj²†jI⁰…z†j …1 ÿ jI…z†j²†² ˆ 0

(that is, as in Theorem 1 with f…t† ˆ t²). Let J be a measurable subset of T and set

E ˆ I^ÿ1…J †:

Then the function

F…x† ˆ Z _x

0 x_E…e^{i t}† dt belongs to l^…R†.

LoÈwner's lemma asserts, with the above notation, that jEj ˆ jJ j whenever I…0† ˆ 0 and so, for any inner function I, 0 < jEj < 1 if 0 < jJ j < 1. The conclusion of Corollary 3 was considered in [12] where it was shown that if F 2 l^…R† then jEj ˆ 0 or jEj ˆ 1 or dim…¶E† ˆ 1. Thus, if I is as in Corollary 3, the boundary of the pre-image by I of any Borel set of positive measure has Hausdorff dimension 1. In this sense, the inner function I has very wild behaviour.

The proof of Theorem 1 follows from the following two theorems.

Theorem 2. Let f: …0; 1Š ! …0; 1† be a continuous function, f…0^‡† ˆ 0. Then there exists an interpolating Blaschke product B such that

…1 ÿ jzj²†jB⁰…z†j < f…1 ÿ jB…z†j²† for all z 2 D.

Recall that a Blaschke product is called interpolating if inf_n …1 ÿ jznj²†jB⁰…zn†j > 0;

where fz_ng is the zero sequence of B. Such a function cannot belong to B₀ except when it has a ®nite number of zeros.

The function B in Theorem 2 will in fact be a covering map. Theorem 2 permits us to establish Corollaries 1 and 2 with B₀replaced by B, but with the extra conclusion that the corresponding inner function is an interpolating Blaschke product.

Functions in B₀ map hyperbolic discs of a ®xed diameter into euclidean discs of diameter tending to 0 as one approaches T ˆ ¶D. The second step of our construction concerns inner functions which map hyperbolic discs of a ®xed diameter into hyperbolic discs of diameter tending to 0 as one approaches T.

Theorem 3. There exists a non-constant inner function I for which

j z j ! 1lim^ÿ

…1 ÿ jzj²†jI⁰…z†j 1 ÿ jI…z†j² ˆ 0:

…1:1†

Such an inner function I cannot extend analytically to any point of T. Indeed, if I has an angular derivative at the point y 2 T, that is, if the quotient

I…z† ÿ I…y†

z ÿ y

has a limit when z approaches y non-tangentially, then the Julia±CaratheÂodory lemma asserts that

z ! ylim inf …1 ÿ jzj²†jI⁰…z†j 1 ÿ jI…z†j² > 0:

Moreover, although the inner functions of Theorem 3 are in B₀, they form a strict subclass of B₀, because there exist inner functions in B₀ which can be extended analytically to almost every point of T (see for example, [9]). Inner functions in B₀ have been considered by Bishop in [3] and we use some of his ideas.

It is worth mentioning also that the condition (1.1) in Theorem 3 has appeared in [14] in connection with composition operators from B₀ into itself. Indeed, Theorem 3 answers a question in [14, p. 2686] as to whether there is a function f in B0 with C…f† compact as an operator from B0 to B0 such that f…D† Ç T is in®nite. We may take f…z† to be the inner function I…z† of Theorem 3 for which f…D† ˆ D. Also, the completely opposite situation has been considered in [10].

Now suppose that f 2 H¹, with k f k1< 1. For a 2 T the functions H_a…z† ˆa ‡ f …z†

a ÿ f …z†

…1:2†

have positive real part. Hence there exist positive measures ja on T such that the Herglotz representation

Re Ha…z† ˆ Z

TP…z; y† dja…y†

holds for all z 2 D. Here,

P…z; y† ˆ …1 ÿ jzj²†j1 ÿ yzj^ÿ2

denotes the Poisson kernel. It is well known (and easy to prove) that the measure ja is singular for some a 2 T if and only if f is inner. Moreover if f and Ha are related by (1.2) then

j z j ! 1lim^ÿ

…1 ÿ jzj²†j f⁰…z†j 1 ÿ j f …z†j² ˆ 0 if and only if

j z j ! 1lim

…1 ÿ jzj²†jHa⁰…z†j Re H_a…z† ˆ 0:

…1:3†

So to prove Theorem 3 it is suf®cient to construct a singular measure j such that its Herglotz transform H satis®es (1.3).

To avoid endless repetition, J and J⁰ will henceforth, and throughout the paper, denote adjacent arcs of T with jJ j ˆ jJ⁰j.

With this notation we have the following.

Theorem 4. Let H be analytic in D with Re H…z† > 0 for z 2 D. Let j be the corresponding measure on T for which

Re H…z† ˆ Z

TP…z; y† dj…y†:

The following statements are equivalent:

j z j ! 1lim^ÿ

…1 ÿ jzj²†jH⁰…z†j Re H…z† ˆ 0;

…a†

j J j ! 0lim j…J † j…J⁰†ˆ 1:

…b†

Positive measures satisfying (b) are called symmetric (see [8]). Thus, to prove Theorem 3 it is suf®cient to exhibit a positive singular symmetric measure. In fact, such measures were constructed by L. Carleson in [5] in connection with quasiconformal mappings. It is also possible to prove Theorem 3 using a construction of C. Bishop and the following result.

Theorem 5. Given an inner function I, consider the positive measure in D È T,

m ˆ X

z : I…z† ˆ 0

…1 ÿ jzj²†d_z‡ 2j;

where dz denotes the Dirac mass at z, the sum takes into account the multiplicity of the zeros of I, and j is the measure associated with the singular part of I. The following assertions are equivalent:

j z j ! 1lim^ÿ

…1 ÿ jzj²†jI⁰…z†j 1 ÿ jI…z†j² ˆ 0;

…a†

(b) for any « > 0 the following two conditions hold:

d ! 0lim sup

j Q j < d

m…Q†

m…Q⁰†ÿ 1  

: m…Q†

jQj <1

«

 

ˆ 0;

…1:b†

N ! 1lim sup

Q

X¹

k ˆ N

m…2^kQn2^{k ÿ 1}Q†

2^{2 k}m…Q† : m…Q†

jQj <1

«

( )

ˆ 0:

…2:b†

Here Q denotes the Carleson square

Q ˆ fz: z ˆ re^{i v}; v 2 J; 1 ÿ jJ j < jzj < 1g

corresponding to an interval J Ì T, jQj ˆ jJ j and Q⁰ is the corresponding Carleson square for J⁰.

As mentioned above, L. Carleson constructed singular symmetric measures.

Indeed, let w…t† be a continuous increasing function on ‰0; 1Š, with w…0† ˆ 0, such that t^{ÿ1 = 2}w…t† is decreasing. Let j be a positive measure on T such that

jj…J † ÿ j…J⁰†j < w…jJ j†j…J †;

for any arc J of the unit circle. L. Carleson showed that the condition Z

0

w²…t†

t dt < 1

implies that j is absolutely continuous and in fact, its derivative is in L². Conversely, if

Z

0

w²…t†

t dt ˆ 1;

there exists a positive singular measure on T such that jj…J † ÿ j…J⁰†j < w…jJ j†j…J †;

for any arc J of the unit circle.

A similar situation occurs when looking for the best decay one can have in Schwarz's Lemma. Given a positive increasing function w on …0; 1Š, consider

ew…t† ˆ t Z ₁

t

w…s†

s² ds ‡ tw…1† for t 2 …0; 1Š:

…1:4†

Observe that ew…t† > w…t† for 0 < t < 1, and ew…t† < c…«†w…t† if w…t†=t^{1 ÿ «} is decreasing for some « > 0.

Theorem 6. Let w be a positive continuous function on …0; 1Š.

(a) Assume that

Z

0

w²…t†

t dt < 1:

Then there is no non-constant inner function I such that …1 ÿ jzj²† jI⁰…z†j

1 ÿ jI…z†j²< w…1 ÿ jzj†;

for all z 2 D.

(b) Let w be increasing. Assume that there exist constants k and d such that ew…t† < kw…t† if 0 < t < d;

and Z

0

w²…t†

t dt ˆ 1:

Then, there exists a non-constant inner function such that …1 ÿ jzj²† jI⁰…z†j

1 ÿ jI…z†j²< Cw…1 ÿ jzj† for z 2 D;

where C is an absolute constant.

For instance, the function w…t† ˆ j log tj^ÿa satis®es (a) when a >¹₂ and (b) when a <¹₂. The construction of the inner function in part (b) of Theorem 6 uses symmetric singular measures. Actually, we need a re®nement of the Carleson result, where we assume the integral condition and that w…t†=t decreases. This is done in § 6 by means of Riesz products.

Using Theorem 6, one can prove versions of Corollaries 1 and 2 with B₀ replaced by the space B₀…w† of holomorphic functions f in the unit disc such that

j z j ! 1lim^ÿ

…1 ÿ jzj²†j f⁰…z†j w…1 ÿ jzj† ˆ 0;

where w ful®lls the conditions in part (b) of Theorem 6.

Corresponding to the Zygmund class and the Bloch space, there are the Zygmund measures, that is, positive measures m in T for which

jm…J † ÿ m…J⁰†j ˆ O…jJ j† as jJ j ! 0:

This condition is equivalent to the fact that the primitive of m is in the Zygmund class. Piranian [17] and Kahane [13] constructed ®nite positive singular measures satisfying

jm…J † ÿ m…J⁰†j ˆ o…jJ j† as jJ j ! 0:

We call such measures Kahane measures. Using Theorem 1 or Theorem 6 we will construct measures which are simultaneously symmetric and Kahane. In fact, as is to be expected from [5] and [13], one is able to replace the o…1† condition by a condition of the form O…w…jJ j††, where w ful®lls the conditions in part (b) of Theorem 6. The point is that we do this in a new and uniform way. In private communications, A. Canton [4] and F. Nazarov showed us other ways of producing Kahane symmetric measures.

Also, one can establish the following sharp version of Corollary 3.

Corollary 4. Let a be a positive increasing function on …0; 1Š, with a…0^‡† ˆ 0. Assume that a…t†=t^{1 ÿ «} is decreasing for some « > 0. Then, the following assertions are equivalent:

(a) there exists a measurable set E Ì T, with 0 < jEj < 1, such that the measure x_Ejdyj is a-symmetric, that is,

j jE Ç J j ÿ jE Ç J⁰j j < a…jJ j†jE Ç J j;

for any arc J Ì T;

(b) there exists a measurable set E Ì T, with 0 < jEj < 1, such that the measure x_Ejdyj is a-Zygmund, that is,

j jE Ç J j ÿ jE Ç J⁰j j < a…jJ j†jJ j;

for any arc J Ì T;

Z

0

a²…t†

t dt ˆ 1:

…c†

The hyperbolic metric in D is the Riemannian metric l_D…z† jdzj, where l_D…z† ˆ …1 ÿ jzj²†^ÿ1. Let Q be a hyperbolic domain, that is, a domain in the complex plane whose complement has at least two points. Let p: D ! Q be a universal covering map. Then l_D projects to the Poincare metric l_Q…z† jdzj of Q, where

l_Q…p…z†† ´ jp⁰…z†j ˆ l_D…z†:

Schwarz's lemma asserts that holomorphic mappings f from D into Q decrease hyperbolic distances, or in®nitesimally,

…1 ÿ jzj²†j f⁰…z†jl_Q… f …z†† < 1;

for all z 2 D.

A holomorphic function f from the unit disc into Q is called inner (into Q) if fe^{i v}: lim

r ! 1f …re^{i v}† exists and belongs to Qg ˆ 0:

If p is a holomorphic covering map from D into Q, then p is inner; and as a matter of fact, if f is any holomorphic function from D into Q which factorizes f ˆ p ± b, where b: D ! D, then f is inner (into Q) if and only if b is inner into D (see [7]).

The theorems stated in this introduction have counterparts in this more general setting. For instance, Theorem 6 shows that if Q is a hyperbolic domain and a positive weight satis®es

Z

0

w²…t†

t dt < 1;

then there is no non-constant inner function I into Q such that …1 ÿ jzj²†jI⁰…z†jl_Q…I…z†† < w…1 ÿ jzj†;

for all z 2 D. On the other hand, if w ful®lls the conditions in part (b) of Theorem 6, there exists a non-constant inner function I into Q such that

…1 ÿ jzj²†jI⁰…z†jl_Q…I…z†† < w…1 ÿ jzj† for z 2 D:

The paper is organized as follows. In § 2 we prove Theorem 2 and apply it to establish some results on composition operators. Section 3 contains two proofs of Theorem 3, using Theorems 4 and 5 respectively. Then we use Theorem 3 to establish Theorem 1 and the corollaries mentioned in this introduction, together with other related results. The proof of Theorem 4 is in § 4 and consists of a discretization procedure, which can be adapted to prove Theorem 5. As mentioned, this uses some of the ideas of [3]. In § 5 we prove Theorem 6. This uses the existence of singular symmetric measures proved by L. Carleson and a re®nement of Theorem 4, whose proof is different from the one in § 4. Also, several ways of constructing singular measures which are both symmetric and Kahane are mentioned. Finally in § 6, we construct singular symmetric measures using Riesz products.

After this paper was completed, we learned that Wayne Smith had previously obtained Theorem 6, and hence Theorem 3, by different methods [19].

2. Interpolating Blaschke products and composition operators

The proof of Theorem 2 is based on an estimate of the density of the hyperbolic metric on plane domains, due to Beardon and Pommerenke [2]. We require only a crude estimate of this type, for which we present a proof.

Lemma 2.1. Let Q be a domain in D and let f be an analytic function in D with f …D† Ì Q. Then, for all z 2 D,

…1 ÿ jzj²†j f⁰…z†j < 6 dist… f …z†; ¶Q† log e dist… f …z†; ¶Q†:

Proof. Let a 2 ¶Q be such that dist… f …z†; ¶Q† ˆ j f …z† ÿ aj, and assume ®rst that j f …z† ÿ aj >¹₄…1 ÿ j f …z†j²†:

Then

…1 ÿ jzj²†j f⁰…z†j < 1 ÿ j f …z†j²< 4j f …z† ÿ aj < 6j f …z† ÿ aj log e j f …z† ÿ aj:

If, on the other hand,

j f …z† ÿ aj <¹₄…1 ÿ j f …z†j²† …2:1†

then a 2 D, that is, a 62 T. Since

S…z† ˆ exp ÿ1 ‡ z 1 ÿ z

 

is a universal covering map of the punctured unit disc Dnf0g, there exists a holomorphic mapping f from D into D satisfying

f ÿ a

1 ÿ af ˆ S ± f:

A simple calculation shows that

…1 ÿ jwj²†jS⁰…w†j ˆ 2jS…w†j log jS…w†j^ÿ1 for w 2 D and hence

…1 ÿ jzj²†…1 ÿ jaj²†

j1 ÿ af …z†j² j f⁰…z†j < …1 ÿ jf…z†j²†jS⁰…f…z††j

ˆ 2 f …z† ÿ a 1 ÿ af …z†

log f …z† ÿ a 1 ÿ af …z†

  ^ÿ1: Thus

…1 ÿ jzj²†j f⁰…z†j < 2j1 ÿ af …z†j

1 ÿ jaj² j f …z† ÿ aj log e j f …z† ÿ aj and the result follows from (2.1).

We also use the following elementary result, whose proof is omitted.

Lemma 2.2. Let h: …0; 1Š ! …0; 1Š be a continuous function. Then there exists a countable set L Ì Dnf0g whose cluster set is contained in T such that, for all z 2 D,

dist…z; L È T† < h…1 ÿ jzj†:

Proof of Theorem 2. Given f…t†, consider a continuous function h: …0; 1Š ! …0; 1Š

satisfying

6h…t† log e

h…t†< f…t†

for all t 2 …0; 1Š. For the set L of Lemma 2.2, let B be a holomorphic universal covering of D onto Q ˆ DnL. Then Lemmas 2.1 and 2.2 show that

…1 ÿ jzj²†jB⁰…z†j < f…1 ÿ jB…z†j²†;

as required and it remains to show that B is an interpolating Blaschke product.

Since B 2 H¹, its radial limit B…y† exists for almost every y 2 T. Moreover, since B is a covering, B…y† 2 L È T and hence in fact B…y† 2 T for almost every y 2 T since L is countable. Thus B is inner.

If B had a singular inner factor then there would be at least one value of y 2 T,

y0 say, with

r ! 1lim^ÿ B…ry0† ˆ 0:

We have arranged that 0 62 L and so this cannot happen. Thus B is a Blaschke product. To prove that it is interpolating it is suf®cient to observe that the quantity …1 ÿ jzj²†jB⁰…z†j depends only on B…z†. Indeed, if B…a† ˆ B…b†, then there exists an automorphism f of D such that f…a† ˆ b and B ± f  B. Hence

…1 ÿ jbj²†jB⁰…b†j ˆ …1 ÿ jaj²†jf⁰…a†j jB⁰…b†j ˆ …1 ÿ jaj²†jB⁰…a†j:

Thus

infn f…1 ÿ jz_nj²†jB⁰…z_n†j: B…z_n† ˆ 0g > d > 0 for some suitable d as required.

Remarks. 1. There exists also a singular inner function satisfying Theorem 2.

In fact we may take a universal covering map of Q È f0g. Such a function will again not belong to B₀.

2. By taking the set L as close to the unit circle as we please, we can have inff…1 ÿ jzj²†jB⁰…z†j: B…z† ˆ 0g > 1 ÿ d

for any preassigned d > 0, even though Schwarz's Lemma tells us that …1 ÿ jzj²†jB⁰…z†j < 1

for all z 2 D. Actually, if the covering map B satis®es B…D† É rD, with 0 < r < 1, one has …1 ÿ jzj²†jB⁰…z†j > r, provided B…z† ˆ 0.

Now suppose that B 2 H¹ with kBk1< 1. It was shown in [14] that the composition operator C…B† is compact in B if and only if

…1 ÿ jzj²†jB⁰…z†j ˆ o…1†…1 ÿ jB…z†j²† as jB…z†j ! 1:

Thus Theorem 2 has the following corollary.

Corollary 2.3. There exists an interpolating Blaschke product B such that the composition operator

C…B†: B ! B; C…B†… f † ˆ f ± B is compact.

Next we consider the space H…w† of analytic functions in the unit disc such that the norm

k f k_w ˆ sup j f …z†j w…jzj†: z 2 D

 

< 1:

Here w denotes a positive continuous function on ‰0; 1† with lim_{t ! 1}^ÿw…t† ˆ 1.

Corollary 2.4. For any function w as above and « > 0, there exists an interpolating Blaschke product B such that the composition operator C…B† maps H…w† into the Bloch space B and

kC…B†… f †k_B< «k f k_w for any f 2 H…w†.

Proof. Replacing w by «^ÿ1w, one can assume that « ˆ 1. If f 2 Hw and k f kwˆ 1 then, from Cauchy's inequality,

…1 ÿ jzj²†j f⁰…z†j < 4w…jzj ‡¹₂…1 ÿ jzj††:

If we choose f…t† so that

w…t ‡¹₂…1 ÿ t††f…1 ÿ t²† < 1

for 0 < t < 1 then f…t† ! 0 as t ! 0. By Theorem 2, there exists an interpolating Blaschke product B such that

…1 ÿ jzj²†jB⁰…z†j < f…1 ÿ jB…z†j²† for z 2 D. Hence for all z 2 D,

…1 ÿ jzj²†j… f ± B†⁰…z†j < 1:

Remarks. 1. The point about the last result is that one inner function suf®ces for all the functions in H…w†. It is easy to see that for any given analytic function f there is an inner function I ˆ I … f † so that f ± I 2 B. Actually one may take I to be the universal covering map of Dnf f^ÿ1…m ‡ ni†: m; n 2 Zg.

2. Elementary considerations enable us to replace the space H…w† by similar spaces de®ned in terms of the growth of derivatives.

3. In the proof of Corollary 2.4 we may choose a function f…t† such that w…t ‡¹₂…1 ÿ t††f…1 ÿ t²† ! 0 as t ! 1^ÿ:

Applying [14, Theorem 2] or Corollary 2.3, one can arrange that the composition operator

C…B†: H…w† ! B is compact.

Corollary 2.5. Given a sequence f fng of functions analytic in D, there exists an interpolating Blaschke product B such that fn± B 2 B for n ˆ 1; 2; 3; . . . :

Proof. It suf®ces to observe that there is a function w…r† such that fn2 H…w†

for n ˆ 1; 2; 3; . . . : For instance, we may take

w…r† ˆ X

n < …1 ÿ r†^ÿ1

supfj fn…z†j: jzj ˆ rg:

Remark. In a way similar to the above, we may replace the sequence f fng by a sequence fAng of Banach spaces of holomorphic functions in D and get the corresponding result that f_n± B 2 B for any f_n2 A_n. Derivatives may be treated similarly, but we omit the details.

Finally in this section, we consider the case f…t† ˆ ct², for c > 0, in Theorem 2; that is, let I be an inner function satisfying

…1 ÿ jzj²†jI⁰…z†j < c…1 ÿ jI…z†j²†²: …2:2†

For any a 2 T consider the holomorphic function Faˆ …a ‡ I †=…a ÿ I †:

Since Re Fa> 0 for z 2 D, there exists a positive measure ja in T such that Re Fa…z† ˆ

Z

T P…z; y† dja…y†

for all z 2 D. Since I is inner, the measures ja are singular and a simple calculation shows that for all a 2 T,

kFakB< 8c:

Thus the measures j_a satisfy the Zygmund condition uniformly in a. In other words, there is a constant C₁ such that

jj_a…J † ÿ j_a…J⁰†j < C₁jJ j for all a 2 T and all J, J⁰.

Denote by A…I † the j-algebra generated by the preimages under I of the Lebesgue measurable sets in T and the sets of measure 0.

Theorem 2.6. Let I be an inner function satisfying (2.2), and let h 2 L¹…T† be measurable with respect to the j-algebra A…I †. Then the Cauchy transform of h, that is,

F…z† ˆ Z

T

h…y† dy

1 ÿ yz for z 2 D;

is in the Bloch space B.

Proof. We claim that for any g 2 L¹…T† and any I inner we have 1

2p Z _2p

0

g…I…e^{i v}††

1 ÿ e^{ÿi v}z dv ˆ X

n < ÿ1

bg…n†…I…0††^ÿn‡ 1 2p

Z _2p

0

g…e^{i v}† dv 1 ÿ e^{ÿi v}I…z†: One proves this for g…y† ˆ yⁿ with n 2 Z, applying Cauchy's formula when n > 0 or the mean value theorem when n < 0.

Now there exists g 2 L¹…T† such that h ˆ g ± I and it suf®ces to show that Z _2p

0

g…e^{i v}†

1 ÿ e^{ÿi v}I…z†dv 2 B:

We observe that the function f …z† ˆ

Z _2p

0

g…e^{i v}† 1 ÿ e^{ÿi v}zdv

belongs to H…w† where w…t† ˆ …1 ÿ t†^ÿ1. If I is an inner function satisfying (2.2) then the proof of Corollary 2.4 shows that f ± I 2 B as required.

The following corollary is now immediate.

Corollary 2.7. Under the assumptions of Theorem 2.6, the function F…x† ˆ

Z _x

0 h…e^{i t}† dt; with h 2 L¹; belongs to the Zygmund class L^…R†.

3. Inner functions in the small hyperbolic Lipschitz space As before we consider the equation

Re Ha…z† ˆ Rea ‡ f …z†

a ÿ f …z†ˆ Z

T P…z; y† dja…y†;

…3:1†

where a 2 T, f 2 H¹ with k f k1< 1 and ja…y† is the associated positive probability measure on T. The function f is inner if and only if the measure ja

is singular for some a 2 T. In particular, if ja is singular for some a 2 T then ja

is singular for all a 2 T. Also, the support of ja is a ®nite set if and only if f is a

®nite Blaschke product. So this condition is also independent of a 2 T. However, the fact that j_a satis®es some property usually does not imply that j_b satis®es the same property if b 6ˆ a. See [1], where some examples are considered.

Nevertheless, the fact that f satis®es the conclusion of Theorem 3 can be rephrased in terms of j_a, with a 2 T.

Proposition 3.1. Suppose that f 2 H¹ with k f k₁< 1. The following assertions are equivalent:

j z j ! 1lim

…1 ÿ jzj²†j f⁰…z†j 1 ÿ j f …z†j² ˆ 0;

…a†

Z

T

y dja…y†

…1 ÿ yz†²

 ˆ o…1†Z

T

dja…y†

j1 ÿ yzj² as jzj ! 1^ÿ; …b†

j z j ! 1lim^ÿ

…1 ÿ jzj²†jH_a⁰…z†j Re Ha…z† ˆ 0;

…c†

where f , H_a and j_a are related by (3.1).

Proof. Fix a 2 T. If H_aˆ …a ‡ f †…a ÿ f †^ÿ1, then f ˆ a…H_aÿ 1†…H_a‡ 1†^ÿ1and 1 ÿ j f j² ˆ 4 Re Ha

j1 ‡ Haj²; f⁰ˆ 2aHa⁰

…Ha‡ 1†²: Thus,

jHa⁰j

Re H_aˆ 2j f⁰j 1 ÿ j f j²: Thus condition (a) may be written as

j z j ! 1lim

…1 ÿ jzj²†jH_a⁰…z†j Re Ha…z† ˆ 0 and since

Ha⁰…z† ˆ 2 Z

T

y dj_a…y†

…1 ÿ yz†²; and

Re H_a…z† ˆ Z

T

…1 ÿ jzj²† dj_a…y†

j1 ÿ yzj² ; the result follows.

The proof of Theorem 3 now follows from Proposition 3.1, Theorem 4 and the existence of singular symmetric measures. We may also prove Theorem 3 from the following proposition.

Proposition 3.2. Let j be a positive measure on T and set S‰jŠ…z† ˆ exp ÿ

Z

T

y ‡ z y ÿ z dj…y†

 

: Then j is symmetric if and only if

j z j ! 1lim^ÿ

…1 ÿ jzj²†jS‰jŠ⁰…z†j jS‰jŠ…z†j log…jS‰jŠ…z†j^ÿ1†ˆ 0:

Proof. If

H…z† ˆ Z

T

y ‡ z

y ÿ zdj…y† for z 2 D;

then

…1 ÿ jzj²†jS‰jŠ⁰…z†j

jS‰jŠ…z†j log…jS‰jŠ…z†j^ÿ1†ˆ…1 ÿ jzj²†jH⁰…z†j Re H…z† ; and the result follows from Theorem 4.

Note that whenever j is a singular symmetric measure, then Theorem 3 holds for I ˆ S‰jŠ.

There is yet another way of proving Theorem 3. In [3], Bishop has constructed a Blaschke product in B₀. In fact, if

m ˆ X

z ; B…z† ˆ 0

…1 ÿ jzj²†d_z;

then his construction satis®es

j Q j ! 0lim m…Q†

m…Q⁰†ˆ 1;

…3:2†

where, as before, Q and Q⁰ are contiguous Carleson squares of the same size.

Applying Theorem 5 one can easily show that (3.2) implies that

j z j ! 1lim

…1 ÿ jzj²†jB⁰…z†j 1 ÿ jB…z†j² ˆ 0:

Observe also that, by Proposition 3.1 and Theorem 4, the corresponding singular measures j_a, with a 2 T, will be symmetric.

The next corollary follows from Theorem 3 and Theorem 1 in [14].

Corollary 3.3. There exists an inner function I such that the composition operator C…I † maps B into B₀ compactly.

Proof of Theorem 1 and Corollaries 1 and 2. We set I…z† ˆ B…I0…z††

where B satis®es the hypotheses of Theorem 2 and I0 the hypotheses of Theorem 3. Then

…1 ÿ jzj²†jI⁰…z†j

f…1 ÿ jI…z†j²† ˆ…1 ÿ jzj²†jB⁰…I₀…z††j jI₀⁰…z†j f…1 ÿ jB…I₀…z††j²†

<…1 ÿ jzj²†jI₀⁰…z†j

1 ÿ jI₀…z†j² ! 0 as jzj ! 1^ÿ:

Corollaries 1 and 2 then follow also from Corollaries 2.4 and 2.5 by composing with the same inner function I0. Observe that in any of these results the inner function whose existence is asserted can be chosen to be singular or a Blaschke product. Moreover Remarks 2 and 3 after Corollary 2.4 and the Remark after Corollary 2.5 apply with B replaced by B0.

Ideals in the space of inner functions

Let D be the set of inner functions I for which

j z j ! 1lim^ÿ

…1 ÿ jzj²†jI⁰…z†j 1 ÿ jI…z†j² ˆ 0:

We note that D is an ideal in the space of inner functions with respect to composition from the left. In fact, if I 2 D and f 2 H¹ with kfk₁< 1 then it follows from Schwarz's lemma that

…1 ÿ jzj²†jf⁰…I…z††j jI⁰…z†j

1 ÿ jf…I…z††j² <…1 ÿ jzj²†jI⁰…z†j 1 ÿ jI…z†j² :

This shows again that the inner function in Theorem 3 can be taken to be a singular inner function as well as a Blaschke product.

The next result asserts that the only primary ideals (with respect to left composition) of inner functions contained in B₀ are the ones given by functions in D.

Proposition 3.4. Let I be an inner function such that f ± I 2 B0 for any inner function f. Then I 2 D.

Proof. It is obvious that I 2 B0. If I 62 D then there exists fzng Ì D such that

n ! 1lim jI…z_n†j ˆ 1 and

…1 ÿ jznj²†jI⁰…zn†j

1 ÿ jI…zn†j² > d > 0

for n ˆ 1; 2; 3; . . . : Passing to a subsequence, if necessary, we may assume that fI…zn†g forms an interpolating sequence for H¹. If f is the corresponding interpolating Blaschke product, then for n ˆ 1; 2; 3; . . . one has

…1 ÿ jI…zn†j²†jf⁰…I…zn††j > C;

and

…1 ÿ jznj²†jI⁰…zn†j jf⁰…I…zn††j > C…1 ÿ jz_nj²†jI⁰…z_n†j 1 ÿ jI…z_n†j² > Cd;

contradicting the fact that f ± I 2 B₀.

It is worth mentioning that there are no ideals with respect to composition from the right contained in B0. Indeed if one considers the singular inner function

f…z† ˆ exp ÿ 1 ‡ z 1 ÿ z

 

 

;

then I ± f does not belong to B₀ for any non-constant analytic function I. In fact, if z ! 1 along a suitable horocycle then the quantity

…1 ÿ jzj²†jI⁰…f…z††j jf⁰…z†j cannot tend to zero, no matter what I is.

However, there do exist non-trivial right ideals. For instance, if a > 0 then the set

D_aˆ



f : f inner, …1 ÿ jzj²†j f⁰…z†j

…1 ÿ j f …z†j²†^{a ‡ 1}ˆ O…1† as jzj ! 1



is a bilateral ideal. It is interesting to observe that if f 2 D_a and g 2 D_b then f ± g 2 D_{a ‡ b}.

Let us next consider f…t† ˆ t²in Theorem 1 so that I is an inner function satisfying

j z j ! 1lim

…1 ÿ jzj²†jI⁰…z†j …1 ÿ jI…z†j²†² ˆ 0:

…3:3†

Theorem 3.5. Let I be an inner function satisfying (3.3) and let ja, for a 2 T, be the corresponding singular measures de®ned by (3.1). Then ja are (uniformly in a 2 T) Kahane measures, that is,

j J j ! 0lim 1

jJ j…ja…J † ÿ ja…J⁰†† ˆ 0 uniformly for a 2 T.

Proof. It is well known that the Herglotz integral of a positive measure is in B if and only if the measure is Zygmund, and it is in B0 if and only if the measure is in the small Zygmund class (see [18, p. 156]). So it is suf®cient to observe that the functions …a ‡ I †…a ÿ I †^ÿ1 are in B₀ and

supa sup

1 > j z j > 1 ÿ r…1 ÿ jzj²† a ‡ I a ÿ I

 ₀

 …z†

 ! 0 as r ! 1:

Observe that Proposition 3.1 and Theorem 4 also show that j_a are (uniformly in a 2 T) symmetric measures.

The following theorem, whose proof is omitted, is established in a similar manner to Theorem 2.6 and Corollary 2.7. Recall that given an inner function I, A…I † denotes the j-algebra generated by the preimages under I of the Lebesgue measurable sets in T and the sets of measure 0.

Theorem 3.6. Let I be an inner function satisfying (3.3) and let f 2 L¹…T† be measurable with respect to the j-algebra A…I †. Then

(a) the function

G…z† ˆ Z

T

f …y† dy 1 ÿ yz

belongs to B0, and (b) the function

F…x† ˆ Z _x

0 f …e^{i t}† dt belongs to l^…R†.

If one chooses f as the characteristic function of I^ÿ1…J †, one obtains Corollary 3 of § 1.

4. Proofs of Theorems 4 and 5 To prove Theorem 4 we restate condition (a) as

Z

T P…z; y† dj…y†

t…z; y†

 ˆ o…1†Z

T P…z; y† dj…y† as jzj ! 1^ÿ …4:1†

where

t…z; y† ˆ y ÿ z

1 ÿ zy …y 2 T†:

It is readily shown that this is equivalent to (a).

Given a point z ˆ re^{i v}2 D, denote by J…z† the arc of T with centre e^{i v} and (normalized) length 1 ÿ r. Also, given an arc J Ì T and M > 0 let MJ be the arc of the same centre and with jMJ j ˆ M jJ j.

Part I: (b) ) (a). Assume that (b) holds. We ®rst prove the following.

Lemma 4.1. Given « > 0 there exist N > 0 and d > 0 such that if 1 ÿ d < jzj < 1, then

Z

T n NJ…z†P…z; y† dj…y† < « Z

TP…z; y† dj…y†:

The lemma states, roughly speaking, that contributions to the Poisson integral from far away do not matter.

Proof. Given « > 0, choose d so that if J is an arc of T with jJ j < d then jj…J† ÿ j…J⁰†j < «j…J†

and hence

jj…J È J⁰† ÿ 2j…J †j < «j…J †:

Hence, if 2^kjJ j < d, we have

j…2^kJ† < …2 ‡ «†^kj…J †:

We break the integral on the left into dyadic pieces. Let M denote the integer part of log₂…d=…1 ÿ jzj††, so that 2^M…1 ÿ jzj† , d. Then, using crude estimates we obtain

Z

T n NJ…z†P…z; y† dj…y† < C X^M

k ˆ log2N

j…2^kJ…z††

2^2k…1 ÿ jzj†‡X

k > M

j…2^kJ…z††

2^2k…1 ÿ jzj†

!

; where C is an absolute constant.

The ®rst sum is bounded by j…J…z††

jJ…z†j X¹

k ˆ log₂N

2 ‡ « 4

 _k

< « Z

TP…z; y† dj…y†;

if N is suf®ciently large.

Observe now that, for any « > 0, j…J†

jJ j² > 4 2 ‡ «

 

m…2J†

j2J j²

if jJ j is suf®ciently small. Iterating this inequality, we obtain j…J†

jJ j² > C 4 2 ‡ «

 _n

! 1 as n ! 1:

Thus

j J j ! 0lim j…J †

jJ j² ˆ 1:

…4:2†

The second sum above can be estimated by 2j…T†

2^2M4…1 ÿ jzj†,j…T†

d² …1 ÿ jzj†

and from (4.2) if 1 ÿ jzj is suf®ciently small, this does not exceed

«j…J…z††

1 ÿ jzj< « Z

TP…z; y† dj…y†;

as required.

Now let l > 0 be a small number to be ®xed later and divide NJ…z† into N =l arcs each of length l…1 ÿ jzj†. Call these arcs J_k and let the centre of each arc be y_kˆ e^{i vk}. Then

Z

Jk

P…z; y† dj…y†

t…z; y†ÿ P…z; y_k† j…J_k† t…z; yk†

 < …1 ÿ jzj²† Z

Jk

y

…y ÿ z†²ÿ y_k …y_kÿ z†²

dj…y†

< …1 ÿ jzj²† Z

Jk

jy ÿ ykjjyykÿ z²j jy ÿ zj²jykÿ zj² dj…y†

< 4l Z

Jk

P…z; y† dj…y†;

since jy ÿ y_kj < l…1 ÿ jzj† and jyy_kÿ z²j , jy_kÿ zj. Now sum over k to obtain Z

NJ…z† P…z; y† dj…y†

t…z; y†ÿX^{N = l}

k ˆ 1

P…z; y_k† j…Jk† t…z; y_k†

 < 4lZ

TP…z; y† dj…y†:

The estimate (4.1) follows on taking l such that 4l < « provided that we can show that

X^{N = l}

k ˆ 1

P…z; yk† j…J_k† t…z; yk†

 <1 N

Z

T P…z; y† dj…y†

…4:3†

for any z 2 D such that jzj is close enough to 1.

The number of arcs Jk is large but independent of z. Hence if jzj is close enough to 1, we have

jj…Jk† ÿ j…Ji†j < «

2pj…Jk†; for 1 < k; j < N =l:

We write X^{N = l}

k ˆ 1

P…z; yk† j…J_k†

t…z; y_k†ˆX^{N = l}

k ˆ 1

P…z; yk†j…J_k† ÿ j…J₁†

t…z; y_k† ‡ j…J1†X^{N = l}

k ˆ 1

P…z; y_k† t…z; y_k†

ˆ T₁‡ T₂; say.Now

jT₁j < «j…J₁† jJ1j < C«

Z

TP…z; y† dj…y†;

where C is an absolute constant, while T₂ˆj…J1†

jJ₁j X^{N = l}

k ˆ 1

1 ÿ jzj² …ykÿ z†²y_kjJ_kj

since jJ_kj ˆ jJ₁j for 1 < k < N =l. The sum above is a Riemann sum of the integral Z

NJ…z†

1 ÿ jzj² …y ÿ z†² dy;

which an easy calculation shows to be bounded by 1=N. The estimate (4.3) follows on taking N large enough since

j…J1†

jJ₁j < 2j…J…z††

jJ…z†j < C Z

T P…z; y† dj…y†;

where C is an absolute constant.

Part II: (a) ) (b). The proof follows closely the arguments of [3]. Consider the pseudohyperbolic disc centred at z of radius c < 1, that is,

fw: r…w; z† < c < 1g where r…w; z† ˆ w ÿ z 1 ÿ zw  

: Integrate (a) from z to w to obtain, for all c < 1,

r…w; z† < csup

jRe H…w† ÿ Re H…z†j

Re H…z† ! 0 as jzj ! 1:

Thus there exists a function a…r† such that …a† a…r† ! 1 as r ! 1;

…b† sup jRe H…w† ÿ Re H…z†j

Re H…z† : r…w; z† < a…jzj†

 

! 0 as jzj ! 1:

…4:4†

Lemma 4.2. Suppose that (a) holds. Then, given N > 1 there exists d ˆ d…N † 2 …0; 1† such that if 1 ÿ d < jzj < 1, then

Z

T n NJ…z†P…z; y† dj…y† <C N

Z

TP…z; y† dj…y†;

where C is an absolute constant.

Proof. Let d ˆ d…N † be a small number, to be ®xed later, with d < 1=N.

Given z 2 D, with 1 ÿ jzj < d, consider the point zN ˆ …1 ÿ N…1 ÿ jzj††…z=jzj†:

So, J…z_N†  NJ…z† and for y 62 NJ…z† we have jy ÿ z_Nj < C₀jy ÿ zj;

where C₀ is an absolute constant. Hence

P…z_N; y† > C₀^ÿ2NP…z; y†

for y 62 NJ…z†.

Now, if d > 0 is suf®ciently small and 1 ÿ d < jzj < 1, we have Re H…z† >¹₂Re H…z_N†

and hence Z

T P…z; y† dj…y† ˆ Re H…z† >¹₂Re H…z_N† >¹₂C₀^ÿ2N Z

T n NJ…z†P…z; y† dj…y†:

Lemma 4.3. With the above notation, j…J…z††

jJ…z†j ÿ Re H…z†

 ˆ o…1†ReH…z† as jzj ! 1^ÿ: Proof. For a given z 2 D, consider the arc

L ˆ fre^iv: jv ÿ arg zj < p…1 ÿ d†…1 ÿ jzj†g where r ˆ r…z†, d ˆ d…z† will be chosen later to satisfy

r ! 1; d ! 0; 1 ÿ r

…1 ÿ jzj†d! 0; as jzj ! 1^ÿ: Given « > 0, Lemma 4.2 shows that, for any w 2 L,

Re H…w† ÿ Z

J…z†P…w; y† dj…y†

 < «ReH…z†

provided that …1 ÿ r†=d…1 ÿ jzj† is suf®ciently small. Thus

w 2 Lsup 1

Re H…z† Re H…z† ÿ Z

J…z† P…w; y† dj…y†

 ! 0 as jzj ! 1^ÿ: Integrating along the arc L we obtain

jLj Re H…z† ÿ 1 2p

Z

J…z†

Z

L P…w; y† dj…y† jdwj

 ˆ o…1†jLjReH…z† as jzj ! 1^ÿ:

Now jJ…z†j ÿ jLj ˆ d…1 ÿ jzj† ! 0 and 1

2p Z

L P…w; y† jdwj ! 1 as jzj ! 1^ÿ

if jv ÿ arg zj < p…1 ÿ c†…1 ÿ jzj†. This shows that for any small number c > 0, we have lim inf

j z j ! 1

j…J…z††

jJ…z†j Re H…z†> 1 ÿ c and

lim sup

j z j ! 1

j……1 ÿ c†J…z††

jJ…z†j Re H…z†< 1 ÿ c:

Consider the point w such that J…w† ˆ …1 ÿ c†J…z†, that is, w ˆ …1 ÿ …1 ÿ c†…1 ÿ jzj††…z=jzj†:

The second inequality gives lim sup

j w j ! 1

j…J…w††

…1 ÿ c†^ÿ1jJ…w†j Re H…w†< 1 ÿ c:

Thus,

1 ÿ c < lim inf

j z j ! 1

j…J…z††

jJ…z†j Re H…z†< lim sup

j z j ! 1

j…J…z††

jJ…z†j Re H…z†< 1;

for any small number c > 0. This proves the lemma.

The proof that (a) ) (b) now follows immediately. For contiguous arcs J, J⁰ with centres z and z⁰ (and, as always, the same length),

j…J †

jJ j ÿj…J⁰† jJ⁰j

 < j…J†

jJ j ÿ Re H…z†

 ‡ j…J⁰†

jJ⁰j ÿ Re H…z⁰†

‡ jRe H…z† ÿ Re H…z⁰†j:

Lemma 4.3 shows that the ®rst two terms are bounded by «…Re H…z† ‡ Re H…z⁰††.

Also z and z⁰ are within a bounded hyperbolic distance of each other and hence by (4.4) the last term is also less than «…Re H…z††. Summing up, we have

j…J†

jJ j ÿj…J⁰† jJ⁰j

 < 4«ReH…z† < 5«j…J † jJ j ; as required.

A little consideration shows that the proof of Theorem 4 may be applied to prove the following more general result.

Theorem 4.4. Let f fz: z 2 Dg be a family of positive continuous functions on T. Assume that there exist constants C; M > 0 such that for all z 2 D and all y1; y22 T we have

M^ÿ1< fz…y1† < M;

j f_z…y₁† ÿ f_z…y₂†j < C

1 ÿ jzjjy₁ÿ y₂j:

Assume, further, that j is a symmetric measure on T. Then (4.5)

j z j ! 1lim

1 j…J…z††

Z

T f_z…y†P…z; y† dj…y†

 

1 jJ…z†j

Z

T f_z…y†P…z; y†jdyj 2p

 

 

ˆ 1:

Proof. (This is merely sketched.) As in Lemma 4.1 one may replace the integrals in (4.5) by integrals on NJ…z† for large N. The Riemann sum argument used to prove that (b) ) (a) can now be applied.

Corollary 4.5. Let j be a symmetric measure on T and suppose that f is a continuous function on T. Then

j z j ! 1lim^ÿ

1 ÿ jzj j…J…z††

Z

T… f ± t_z†…y†P…z; y† dj…y† ˆ Z

T f …y†jdyj 2p where, as before,

t_z…y† ˆ y ÿ z 1 ÿ zy:

Proof. Theorem 4.4 can be applied directly if the continuous function satis®es a Lipschitz condition,

j f …y₁† ÿ f …y₂†j < Cjy₁ÿ y₂j on T. Moreover for f  1 one obtains

j z j ! 1lim

1 ÿ jzj j…J…z††

Z

T P…z; y† dj…y† ˆ 1:

…4:6†

Consequently,

z 2 Dsup

1 ÿ jzj j…J…z††

Z

TP…z; y† dj…y† < 1:

Applying the Banach±Steinhaus theorem, we obtain the desired equality for any continuous function f .

Corollary 4.6. Let j be a symmetric measure on T and f be a continuous function on T. Then

j z j ! 1lim R

T… f ± tR z†…y†P…z; y† dj…y†

TP…z; y† dj…y† ˆ Z

T f …y†jdyj 2p : Proof. It suf®ces to apply (4.6) and Corollary 4.5.

Observe that by taking f …z† ˆ z, this corollary proves (b) ) (a) in Theorem 4.

Proof of Theorem 5. This is similar to that of Theorem 4 and so is only sketched.

Part I: (b) ) (a). Using the characterization of the inner functions in B0 given by Bishop in [3] one can easily see that I 2 B0. Hence in proving (a) one may assume that jI…z†j >¹₂. A computation with logarithmic derivatives shows that

…1 ÿ jzj²†jI⁰…z†j ˆ jI…z†j Z

DP…z; y†dm…y†

t…z; y†

; …4:7†

while

1 ÿ jI…z†j², log jI…z†j^ÿ2, Z

DP…z; y† dm…y†

and it is these last two integrals which one has to compare.

For ®xed h > 0, condition (2.b) of Theorem 5 yields an N > 0 such that Z

D n NQ…z†P…z; y† dm…y† < h Z

DP…z; y† dm…y†;

if jzj is suf®ciently close to 1. For such a z consider the ‰N =hŠ disjoint Carleson squares, Q_k say, with k ˆ 1; 2; . . . ; ‰N =hŠ, of size h…1 ÿ jzj† contained in NQ…z†.

Since I 2 B₀ and jI…z†j >¹₂, the zeros of I are (hyperbolically) distant from z and we can assume that the zeros of I in NQ…z† are contained in S

kQ_k. Thus m…NQ…z†† ˆ m [

k

Qk

 :

As in the previous proof, the principal idea is to discretize the integral in (4.7) and compare it with an integral with respect to Lebesgue measure. If we write A , B to mean

jA ÿ Bj < h Z

DP…z; y† dm…y†;

then given points y_k2 Q_kÇ T, one can show, as before, that X

k

Z

Qk

P…z; y† dm…y†

t…z; y†,X

k

P…z; y_k† m…Qk† t…z; y_k† ,m…Q…z††

jQ…z†j X

k

1 ÿ jzj²

…y_kÿ z†…1 ÿ zy_k†jQ_kj

using (1.b) of Theorem 5 in the second estimate. Finally, one only has to observe that the last sum is a Riemann sum for the integral

Z

NQ…z† Ç T

1 ÿ jzj² …y ÿ z†² dy and that this is bounded by 1=N.

Part II: (a) ) (b). As in the proof of Theorem 4, one can show that, given h > 0, there exist N > 0 and d > 0 such that

Z

D n NQ…z†P…z; y† dm…y† < h Z

DP…z; y† dm…y†

…4:8†

if 0 < 1 ÿ jzj < d. To prove (1.b) of Theorem 5, it is suf®cient to show that, for any « > 0,

z: j I…z†j > «sup

j…m…Q…z††=jQ…z†j† ÿ log jI…z†j^ÿ1j

log jI…z†j^ÿ1 ! 0 as jzj ! 1^ÿ: …4:9†

The estimate (4.9) can be proved with the same integration technique used in the corresponding implication in Theorem 4. Finally, to prove (2.b) of Theorem 5

we use (4.8) and (4.9) to show that Z

D n NQ…z†P…z; y† dm…y† < 2hm…Q…z††

jQ…z†j

if m…Q…z†† > «jQ…z†j. One now estimates the left-hand side dyadically to obtain (2.b). The details are omitted.

5. The decay in Schwarz's lemma and symmetric and Kahane measures The existence of the function H…z† of Theorem 4 as well as the existence of the inner function of Theorem 3 both depend ultimately on the existence of singular symmetric measures. In connection with the Beurling±Ahlfors extension theorem for quasi-conformal mappings, L. Carleson has shown [5] that such measures do exist. Indeed if w…t† is a continuous increasing function on ‰0; 1Š with w…0† ˆ 0, such that t^{ÿ1 = 2}w…t† is decreasing and such that

Z

0

w²…t†

t dt ˆ 1;

…5:1†

then there exists a singular measure j on R such that

x 2 Rsup

j…x; x ‡ h†

j…x ÿ h; x†ÿ 1

 < w…h† for h > 0:

…5:2†

Thus choosing, for instance, w…t† ˆ …log…1=t††^ÿa, with a <¹₂, one obtains a singular symmetric measure. The integral condition (5.1) is also necessary for the existence of a singular measure satisfying (5.2), as was also established in [5].

Actually, if j is a measure satisfying (5.2) and Z

0

w²…t†

t dt < 1;

then j is absolutely continuous and its derivative is in L²_loc. A similar situation occurs for inner functions.

Theorem 5.1. Let w be a positive continuous function on …0; 1Š. Assume that Z

0

w²…t†

t dt < 1:

Then, there is no non-constant inner function I such that …1 ÿ jzj²† jI⁰…z†j

1 ÿ jI…z†j²< w…1 ÿ jzj†;

for all z 2 D.

Proof. Assume that such an inner function I exists. Consider a positive singular measure j such that

H…z† ˆ1 ‡ I…z†

1 ÿ I…z†ˆ Z _2p

0

e^{i v}‡ z

e^{i v}ÿ zdj…v† for z 2 D:

Then, for all z 2 D we have

H⁰…z†

H…z† ˆ 2I⁰…z†

1 ÿ I…z†²: So,

…1 ÿ jzj²†jH⁰…z†j²

jH…z†j² <w²…1 ÿ jzj†

1 ÿ jzj² for z 2 D:

Therefore log H is an analytic function whose boundary values are of vanishing mean oscillation (see [9, Chapter VI]). In particular, H belongs to the Hardy space H^p, for any p < 1. Since j is a singular measure, Re H…e^{i v}† ˆ 0 for almost every e^{i v}2 T, and this is a contradiction (see [9, p. 95]).

Observe that the previous argument also shows, assuming the integral condition on w, that the only inner functions I satisfying

…1 ÿ jzj²†jI⁰…z†j < w…1 ÿ jzj²† for z 2 D;

are the ®nite Blaschke products.

The converse of Theorem 5.1 is the following.

Theorem 5.2. Let w be a positive increasing function on …0; 1Š, with w…0^‡† ˆ 0. Assume that there exist constants k and d such that

ew…t† < kw…t† if jtj < d;

where ew…t† is given by (1.4), and that Z

0

w²…t† dt t ˆ 1:

Then, there exists an inner function I such that …1 ÿ jzj²† jI⁰…z†j

1 ÿ jI…z†j²< w…1 ÿ jzj† for z 2 D:

We can then use the composition process. Let f be a positive continuous function with f…0^‡† ˆ 0 as in Theorem 2, and let B0 be the interpolating Blaschke product of Theorem 2.

Theorem 5.3. With w, B0, f and I as above, set B ˆ B0± I. Then …1 ÿ jzj²†jB⁰…z†j

f…1 ÿ jB…z†j²† ˆ o…w…1 ÿ jzj²†† as jzj ! 1^ÿ:

This permits us to establish the analogues of Corollaries 1 and 2 with B0

replaced by

B0…w† ˆ f : f analytic in D; lim

j z j ! 1

…1 ÿ jzj²†j f⁰…z†j w…1 ÿ jzj²† ˆ 0

 

; assuming always that w satis®es the conditions in Theorem 5.2.

As before, the case f…t† ˆ t² in Theorem 5.3 is of special interest. If the inner function B is such that

j z j ! 1lim^ÿ

…1 ÿ jzj²†jB⁰…z†j

…1 ÿ jB…z†j²†²w…1 ÿ jzj²†ˆ 0;

then the corresponding family of positive singular measures ja, with a 2 T, satisfy, uniformly in a, the following two conditions simultaneously:

jja…J † ÿ ja…J⁰†j < w…jJ j†ja…J †;

jja…J † ÿ ja…J⁰†j < w…jJ j†jJ j:

…5:3†

The point is, however, that starting from a given symmetric measure j, a whole family fja: a 2 Tg of singular Kahane symmetric measures, with the additional property that ja and jb are mutually singular if a 6ˆ b, can be obtained.

The condition (5.3) follows from the following re®ned version of (a) ) (b) of Theorem 4.

Theorem 5.4. Let H be analytic in D with Re H…z† > 0 for z 2 D. Let j be the corresponding measure on T for which

Re H…z† ˆ Z

TP…z; y† dj…y†:

Assume that

…1 ÿ jzj²†jH⁰…z†j

Re H…z† < a…1 ÿ jzj†

for all z 2 D, where a is a positive increasing function on …0; pŠ, with a…0^‡† ˆ 0. Then

jj…J † ÿ j…J⁰†j < Ca…pjJ j†j…J †;

for any suf®ciently small arc J of the unit circle.

Proof. We will use the following result due to N. G. Makarov. Given an arc J of the unit circle, denote by zJ the point t…0†, equidistant from the ends of J, where t is the automorphism of the unit disc mapping the arc T Ç fRe z > 0g onto J. Also, denote the domain t…fz 2 D: Re z > 0g† by D…J †.

Lemma [15, p. 6]. Let b be an analytic function in D, and J an arc of T, and assume that

…1 ÿ jzj²†jb⁰…z†j < a for z 2 D…J †;

for some a < 2. Then 1 jJ j

Z

J‰exp…b…y† ÿ b…z_J†† ÿ 1Šjdyj 2p

 < C…a†:

Considering H_r…z† ˆ H…rz†, with r < 1, we may assume that H is analytic in a neighbourhood of the unit disc. Given an arc J of the unit circle, replacing H by H ÿ i Im H…z_J†, we also may assume that H…z_J† > 0. Observe that the function b ˆ log H satis®es

…1 ÿ jzj²†jb⁰…z†j < a…1 ÿ jzj†:

Since 1 ÿ jz_Jj < pjJ j, we obtain 1

jJ j Z

J Re H…y†jdyj

2p ÿ Re H…z_J†

 < Ca…pjJj†ReH…z^J†: